Related papers: Self-similar abelian groups and their centralizers

It is well known that every finite subgroup of automorphism group of polynomial algebra of rank 2 over the field of zero characteristic is conjugated with a subgroup of linear automorphisms. We prove that it is not true for an arbitrary…

A monomial algebra B is defined as a quotient of a polynomial ring by a monomial ideal, which is an ideal generated by a finite set of monomials. In this paper, we determine the automorphism group of a monomial algebra B, under the…

We describe the structure of the automorphism groups of algebras Morita equivalent to the first Weyl algebra $ A_1 $. In particular, we give a geometric presentation for these groups in terms of amalgamated products, using the Bass-Serre…

Let $k$ be an algebraically-closed field, and $B$ a unital, associative $k$-algebra with $n := \dim_kB < \infty$. For each $1 \le m \le n$, the collection of all $m$-dimensional subalgebras of $B$ carries the structure of a projective…

We determine all finite p-groups that admit a faithful, self-similar action on the p-ary rooted tree such that the first level stabilizer is abelian. A group is in this class if and only if it is a split extension of an elementary abelian…

We prove that the automorphism group of an arbitrary non-abelian free group is complete. It generalizes the result by J.Dyer and E.Formanek (1975) stating the completeness of automorphism group of finitely generated free groups. Using the…

Let $G$ be a group and $\varphi \in \Aut(G)$. Then the set $G$ equipped with the binary operation $a*b=\varphi(ab^{-1})b$ gives a quandle structure on $G$, denoted by $\Alex(G, \varphi)$ and called the generalised Alexander quandle. When…

We prove that the automorphism groups of simple polarized abelian varieties of odd prime dimension over finite fields are cyclic, and give a complete list of finite groups that can be realized as such automorphism groups.

A finitely presented, torsion free, abelian-by-cyclic group can always be written as an ascending HNN extension Gamma_M of Z^n, determined by an n x n integer matrix M with det(M) \ne 0. The group Gamma_M is polycyclic if and only if…

The structure of the automorphism group of a simple TAI algebra is studied. In particular, we show that $\frac{\bar{\mrm{Inn}} (A)}{\bar{\mrm{Inn}}_{0} (A)}$ is isomorphic (as a topological group) to an inverse limit of discrete abelian…

We give a classification of maximal elements of the set of finite groups that can be realized as the automorphism groups of polarized abelian threefolds over finite fields.

We complete the study of finite and profinite groups admitting an action by an elementary abelian group under which the centralizers of automorphisms consist of Engel elements. In particular, we prove the following theorems. Let $q$ be a…

We obtain a kind of structure theorem for the automorphism group ${\rm Aut}{\cal A}$ of a unital C$^{*}$-algebra ${\cal A}$. According to it, ${\rm Aut}{\cal A}$ can be regarded as a subgroup of the semi-direct product of direct product…

We introduce a dimension group for a self-similar map as the ${\rm K}_0$-group of the core of the $C^*$-algebra associated with the self-similar map together with the canonical endomorphism. The key step for the computation is an explicit…

We study the groups of rational points of abelian varieties defined over a finite field $ \mathbb{F}_q$ whose endomorphism rings are commutative, or, equivalently, whose isogeny classes are determined by squarefree characteristic…

We define a determinant on the group of automorphisms of non-trivial Severi-Brauer surfaces over a perfect field. Using the generators and relations, we extend this determinant to birational maps between Severi-Brauer surfaces. Using this…

A new infinite series of rational affine algebraic varieties is constructed whose automorphism group contains the automorphism group ${\rm Aut}(F_n)$ of the free group $F_n$ of rank $n$. The automorphism groups of such varieties are…

Consider the algebraic dynamics on a torus T=G_m^n given by a matrix M in GL_n(Z). Assume that the characteristic polynomial of M is prime to all polynomials X^m-1. We show that any finite equivariant map from another algebraic dynamics…

We introduce the notion of self-similarity for compact quantum groups. For a finite set $X$, we introduce a $C^*$-algebra $\mathbb{A}_X$, which is the quantum automorphism group of the infinite homogeneous rooted tree $X^*$. Self-similar…

We study Abelian groups $A$ with centrally essential endomorphism ring $\text{End}\,A$. If $A$ is a such group which is either a torsion group or a non-reduced group, then the ring $\text{End}\,A$ is commutative. We give examples of Abelian…